Modal logics of sabotage revisited

Abstract

Sabotage modal logic was proposed in 2003 as a format for analyzing games that modify graphs they are played on. We investigate some model-theoretic and proof-theoretic aspects of sabotage modal logic, which has come to be viewed as an early dynamic logic of graph change. Our first result is a characterization theorem for sabotage modal logic as a fragment of first-order logic which is invariant with respect to a natural notion of 'sabotage bisimulation'. Next, we offer a sound and complete tableau method and its associated labeled sequent calculus for analyzing reasoning in sabotage modal logic. Finally, we identify and briefly explore a number of open research problems concerning sabotage modal logic that illuminate its complexity, placing it within the current landscape of modal logics that analyze model update, and, returning to the original motivation of sabotage, fixed-point logics for network games.

On the Right Path: A Modal Logic for Supervised Learning

“…Therefore, the proof of the Zig − 1 clause is completed. (5). Similar to (4), we can prove that the Zag − 1 condition is satisfied.…”

“…Relevant and Future Research Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory. We are inspired by the work on SG [19], SML [5] and their application to formal learning theory [12]. This article is also relevant to other work studying graph games with modal logics, such as [9,14,18,20].…”

“…To reason about players' strategies in SLG, in what follows we will study SLG from a modal perspective. Sabotage modal logic (SML) [5] is known to be a suitable tool to characterize SG, which extends basic modal logic with a sabotage modality − ϕ stating that there is an edge such that, ϕ is true at the evaluation node after deleting the edge from the model. However, given the differences between SG and SLG, we will develop a novel modal logic of supervised learning (SLL) to capture SLG.…”

Losing connection: the modal logic of definable link deletion

“…Formula (4) allows us to reduce a formula including [− ] to an L -formula. By (5), when all formulas occurring in [− ] are propositional atoms, the order of different operators [− ] can be interchanged. Actually each propositional atom occurring in formulas (3)-(5) can be replaced by any Boolean formula without affecting their validity.…”

“…But they must be modified, since we have made the obstructing player both less powerful (given the local nature of his choices) and more powerful (since he can remove more than one link in general). More concretely, to analyze the sabotage game, [5] presents a sabotage modal logic (SML) extending standard modal logic with a modality ϕ stating that ϕ is true at the evaluation point after removing some accessibility arrow from the model. But what is a suitable logic for S d G?…”